Built-in Mathematica Symbol

KleinInvariantJ

]
gives the Klein invariant modular elliptic function

.

For certain special arguments, KleinInvariantJ automatically evaluates to exact values.

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

KleinInvariantJ threads element-wise over lists:

Some modular properties of KleinInvariantJ are automatically applied:

Verify a more complicated identity numerically:

Find values at quadratic irrationals:

KleinInvariantJ is a modular function. Make an ansatz for a modular equation:

Form an overdetermined system of equations and solve it:

This is the modular equation of order 2:

Solution of the Chazy equation

:

Plot the solution:

Plot the absolute value in the complex plane:

Plot the imaginary part in the complex plane:

Find derivatives:

Find a numerical root:

Machine-precision input is insufficient to give a correct answer:

With exact input, the answer is correct:

KleinInvariantJ remains unevaluated outside of its domain of analyticity: